Let $n_k(s)$ be the maximal length $n$ such that a quaternary additive $[n,k,n-s]_4$-code exists. We solve a natural asymptotic problem by determining the lim sup $λ_k$ of $n_k(s)/s,$ and the smallest value of $s$ such that $n_k(s)/s=λ_k.$ Our new family of quaternary additive codes has parameters $[4^k-1,k,4^k-4^{k-1}]_4=[2^{2k}-1,k,3\cdot 2^{2k-2}]_4$ (where $k=l/2$ and $l$ is an odd integer). These are constant-weight codes. The binary codes obtained by concatenation meet the Griesmer bound with equality. The proof is in terms of multisets of lines in $PG(l-1,2).$